倒れる "taoreru" is used for objects that fall down while stationary. Therefore, this is used for inanimate objects and for people who collapse.
転ぶ "korobu", on the other hand, is used for objects that fall while moving, so this is used for events like slipping while skiing or skating.
Akira Miura, Essential Japanese Vocabulary, Tuttle
Monday, May 13, 2013
Jim Breen's EDICT doesn't really provide the difference between the three different ways to write "akeru": 開ける、空ける、明ける, so I'm going to write this article for my reference. I hope others will find this useful as well. All definitions come from Kenkyusha's 5th Shinwaei Chuujiten and the Daijiten (研究者新和英中辞典・大辞典) as well.